Unit · year 1
MU-103 · Linear Algebra I
Threads structure6 theorems
Vectors, matrices, and linear maps — the language of everything linear.
Theorems in this unit
T-015
The rank–nullity theorem
For a linear map, rank plus nullity equals the dimension of the domain.
T-016
The invertible matrix theorem
A dozen conditions on a square matrix are all equivalent to invertibility.
T-017
Multiplicativity of the determinant
det(AB) = det(A)det(B).
T-018
The Cauchy–Schwarz inequality
The inner product of two vectors is bounded by the product of their norms.
T-019
The Gram–Schmidt process
Any basis can be turned into an orthonormal one.
T-020
Cramer's rule
Solutions of a square linear system as ratios of determinants.